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Mathematics > Combinatorics

Title:Polynomial expressions of $p$-ary auction functions

Abstract: Let $\mathbb{F}_p$ be the finite field of prime order $p$. For any function
$f \colon \mathbb{F}_p{}^n \to \mathbb{F}_p$, there exists a unique polynomial
over $\mathbb{F}_p$ having degree at most $p-1$ with respect to each variable
which coincides with $f$. We call it the minimal polynomial of $f$. It is in
general a non-trivial task to find a concrete expression of the minimal
polynomial of a given function, which has only been worked out for limited
classes of functions in the literature. In this paper, we study minimal
polynomial expressions of several functions that are closely related to some
practically important procedures such as auction and voting.